报告题目：On Instability and Stability of Gravity Driven Navier-Stokes-Korteweg Model in Two Dimensions

报告时间：2022-11-04  15:00 - 16:00

报告人：江飞 教授  福州大学

腾讯会议ID：953-6730-7604  会议密码：1104

报告链接：https://meeting.tencent.com/dm/ZA93IvnAaFNZ

Abstract：Bresch-Desjardins-Gisclon-Sart have formally derived that the capillarity can slow the growth rate of Rayleigh-Taylor (RT) instability in the capillary fluids based on the linearized two-dimensional (2D) Navier-Stokes-Korteweg equations in 2008. Motivated by their linear theory, we further investigate the nonlinear Rayleigh-Taylor instability problem for the 2D incompressible case in a horizontal slab domain with Navier boundary condition, and rigorously verify that the RT instability can be inhibited by capillarity under our 2D setting. More precisely, if the RT density profile ρ ̅ satisfies an additional stabilizing condition, then there is a threshold of capillarity coefficient κC, such that if the capillary coefficient κ is bigger than κC, then the small perturbation solution around the RT equilibrium state is algebraically stable in time. In particular, if the RT density profile is linear, then the critical number can be given by the formula κC=gh2 /ρ' ̅π2, where g is the gravity constant and h the height of the slab domain. In addition, we also provide a nonlinear instability result for κ∈[0,κC ). The instability result presents that the capillarity cannot inhibit the RT instability, if it's strength is too small. This is a joint work with Fucai Li and Zhipeng Zhang.